Solution to Quiz 1 - First Year Interest Group Seminar | N 1, Quizzes of Health sciences

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Name:

M 408 M

First Day Quiz

1. What is the inverse function of f (x) = ex?

g(x) = ln(x)

2. Choose the correct description(s). A “derivative” is: (a) the slope of a tangent line to a curve taken at a point. (b) an instantaneous rate of change. (c) a limit of difference quotients. (a), (b), and (c) are all correct.
3. Find the limits. Using L’Hospital’s Rule,

a. (^) xlim→ 0

sin(x) x = lim x→ 0

cos(x) 1

= 1

b. lim x→∞

3 x^2 + 4x + 1 9 x^2 − 1000

=

4. Find the derivative of f with respect to x.

f (x) =

√

xln(x)

Using both the chain rule, and the product rule for differentiation,

√ xln(x) =

(xln(x))−^1 /^2 (x

x

• ln(x)) =

√

xln(x)

(1 + ln(x))

5. Determine whether the series converges absolutely, conditionally, or diverges.

∑^ ∞

n=

cos nπ n^3 /^4

Note that since cos(nπ) = ±1, then the series can be expressed

∑^ ∞

n=

(−1)n^

n^3 /^4

The function f (x) = (^) x 31 / 4 is decreasing and lim x→∞ n 31 / 4 = 0. Therefore, the sequence converges conditionally by the alternating series tests. However, the series

∑^ ∞

n=

n^3 /^4

diverges by the p-test, so the convergence is not absolute.

6. Suppose f (x) has a power series representation at a number a. Write down the general formula for the Taylor series of f centered at a.

∑^ ∞

n=

f (n)(a) n! (x − a)n

7. Evaluate the double integral. ∫ (^2)

0

∫ (^) π/ 2

0

x sin y dydx

Working from the inside out, ∫ (^2)

0

∫ (^) π/ 2

0

x sin y dydx =

∫ 2

0

x

∫ (^) π/ 2

0

sin y dydx

=

∫ 2

0

x[− cos y]π/ 0 2 dx

=

∫ 2

0

x[− cos π/2 + cos 0] dx

=

∫ 2

0

x[1] dx

= [x^2 /2]^20 = [2^2 / 2 − 02 /2] = 2